Scale-factor stabilized solid-state laser gyroscope

ABSTRACT

A laser cavity optical architecture of a solid-state laser gyro measures rotational velocity or angular position and is based on a global conservation of the scale factor so that each parameter varies with temperature and avoids optical mode hops.

CROSS REFERENCE TO RELATED APPLICATIONS

The present application is based on International Application No.PCT/EP2005/055035 filed on Oct. 6, 2005, which in turn corresponds toFrance Application No. 0410659 filed on Oct. 8, 2004, and priority ishereby claimed under 35 USC 119 based on these applications. Each ofthese applications are hereby incorporated by reference in theirentirety into the present application.

The field of the invention is that of solid-state laser gyros, used formeasuring rotational velocities or angular positions. This type ofequipment is employed in particular for aeronautical applications.

The laser gyro, developed some thirty years ago, is widely marketed andused today. Its operating principle is based on the Sagnac effect, whichinduces a frequency difference Ω between the two optical emission modespropagating in opposite directions, termed counter-propagating modes, ofa bidirectional ring laser cavity driven in rotation. Conventionally,the frequency difference Ω satisfies:Ω=S· ωwith S=4A/λ·L

where S is called the scale factor of the laser gyro. L and A are theoptical length and the area of the cavity respectively, λ is the averagewavelength of laser emission with no Sagnac effect and ω is the angularrotation velocity of the laser gyro.

The Ω measurement obtained by spectral analysis of the beating of thetwo beams emitted makes it possible to ascertain the value of ω to avery high accuracy.

By electronically counting the beat fringes that move past during achange of angular position, the relative value of the angular positionof the device is also determined.

One of the factors determining the performance of a laser gyro is thetemperature stability of its scale factor S.

In gas lasers of Helium/Neon type comprising a laser cavity and anamplifying medium made up of a gas mixture of helium and neon, thetemperature stability of the scale factor is due to the joint stabilityof the wavelength, of the optical length and of the area of the cavity.Stability of the area of the cavity is obtained by using cavity supportssliced from materials with a very small expansion coefficient such asZerodur. Stability of the laser wavelength is engendered by stability ofthe atomic emission wavelength. The optical length of the cavity dependson the variations in the length and the optical index of the cavity withtemperature. Its stability is ensured by slaving the length of thecavity to the atomic spectral line used. Slaving is carried out by meansof a piezoelectric wedge, the error signal being provided by theintensity of the light emitted by the laser.

The gaseous nature of the amplifying medium is, however, a source oftechnical complications during production of the laser gyro,particularly because of the high gas purity required. Moreover, itcauses premature wear of the laser, this wear being due, in particular,to gas leaks, deterioration of the electrodes and the high voltages usedto establish the population inversion.

Currently, it is possible to produce a solid-state laser gyro thatoperates in the visible or near infra-red by using, for example, anamplifying medium based on neodymium-doped YAG (Yttrium-Aluminum-Garnet)crystals in place of the helium-neon gas mixture, optical pumping thenbeing carried out by laser diodes operating in the near infra-red. Asemiconductor material, a crystalline matrix or a glass doped with ionsbelonging to the class of the rare earths (erbium, ytterbium, etc.) mayalso be used as amplifying medium. All the inherent problems of thegaseous state of the amplifying medium are thus de facto eliminated.

However, in lasers whose amplifying medium is no longer a gas but asolid, the stability of the scale factor cannot be guaranteed by meansof the procedures used for gas lasers.

Specifically, the frequency of the maximum of the amplifying medium gaincurve is subject to sizeable variations as the temperature varies. Forexample, for a laser of neodymium/YAG type, the variation in thefrequency equals −1.3 gigahertz/degree at the wavelength of 1.06 micronsin a temperature range between −50 degrees and +100 degrees Celsius.

It is known that the free spectral interval of a laser cavitycorresponds to the spectral interval separating two frequencies capableof oscillating in the cavity. It equals c/L, c being the velocity oflight in vacuo. In point of fact, in the case of cavities ofconventional size, that is to say for optical lengths L equal to a fewtens of centimeters, and in the case of solid-state lasers, the spectralwidth of the gain curve is large relative to the free spectral interval.Typically, the spectral width of the gain curve represents several tensof free spectral intervals. Under these conditions, it is no longerpossible to construct a slaving of the length of the cavity having thevariations in the gain curve as error signal when the length L varies,as these variations are no longer significant.

Moreover, the variations with temperature of the optical length are muchbigger in solids than in gases. Indeed, the variations in geometriclength due to temperature are compounded with the variations in theoptical index, which are much bigger in a dense medium. Consequently, itis more difficult to compensate for them using a standard piezoelectricwedge.

In order to avoid the need to thermally compensate the scale factor,thermal sensors arranged on the laser cavity could be used to determinethe temperature, then, by virtue of a mathematical model, the variationin the scale factor corresponding to the temperature measured could bededuced therefrom. It would then be possible to introduce this variationinto the calculation of the angular rotation velocity. However,experience proves that models are currently not accurate enough toobtain the desired accuracy.

In the field of solid-state lasers, American patent U.S. Pat. No.6,614,818 proposes an optical architecture that makes it possible topreserve the emission mode without mode hops by globally compensatingfor thermal drifts. This architecture is based on the preservation ofthe emitted mode number n, given by:

$n \approx \frac{v \cdot L}{c}$with v being frequency of the maximum of the gain curve of theamplifying medium used in the laser and L the optical length of thecavity.

To preserve this mode number, it suffices that its variation as afunction of temperature should be zero, which yields the followingmathematical relation:

${{\frac{1}{v} \cdot \frac{\mathbb{d}v}{\mathbb{d}T}} + {\frac{1}{L} \cdot \frac{\mathbb{d}L}{\mathbb{d}T}}} = 0$

This relation could not be applied as it stands to compensate for thevariations in the scale factor, which is a different problem frompreservation of the frequency of the emission mode. In particular, it ispossible to have emission mode hops while still preserving the scalefactor.

The present invention proposes an optical architecture based on globalpreservation of the scale factor, each parameter being able to vary withtemperature.

More precisely, the subject of the invention is a laser gyro comprisingat least one ring cavity of optical length L and of area A and asolid-state amplifying medium that are designed in such a way that twooptical waves of average wavelength λ can propagate in oppositedirections inside the cavity, the scale factor S of the laser gyro beingequal to

$\frac{4 \cdot A}{\lambda \cdot L},$characterized in that the variations in the average wavelength λ, in themagnitude of the area A and in the optical length L of the cavity as afunction of temperature T are such that the scale factor S remainssubstantially constant as the temperature T varies.

Advantageously, as the cavity has a geometric perimeter L_(o), saidcavity comprising at least one optical element Oi, being an integerindex varying between 1 and the total number of optical elements, theoptical element Oi being of length L_(i), of optical index n_(i), x_(i)being equal to the ratio

$\frac{L_{i}}{L_{o}},\frac{\mathbb{d}n_{i}}{\mathbb{d}T}$being the coefficient of variation as a function of temperature T of theoptical index n_(i) of the optical element Oi, α_(i) being the linearexpansion coefficient of the optical element Oi, α_(o) being the linearexpansion coefficient of the material serving as support for the cavity,v being the central emission frequency of the amplifying medium and

$\frac{\mathbb{d}v}{\mathbb{d}T}$being the coefficient of variation as a function of temperature T ofsaid frequency, said optical elements and the amplifying medium aredesigned so that the following relation is substantially satisfied:

${{\left( {{2 \cdot \alpha_{0}} + {\frac{1}{v} \cdot \frac{\mathbb{d}v}{\mathbb{d}T}}} \right)\left\lbrack {1 + {\sum\limits_{i}{\left( {n_{i} - 1} \right)x_{i}}}} \right\rbrack} - \alpha_{0} - {\sum\limits_{i}{\left( {n_{i} - 1} \right){x_{i} \cdot \alpha_{i}}}} - {\sum\limits_{i}{\frac{\mathbb{d}n_{i}}{\mathbb{d}T}x_{i}}}} = 0$

Advantageously, to prevent optical mode hops occurring, the opticalelements and the amplifying medium are designed so that the followingrelations are substantially and simultaneously satisfied:

$\alpha_{0} = {{{{- \frac{1}{v}} \cdot \frac{\mathbb{d}v}{\mathbb{d}T}}\mspace{14mu}{and}\mspace{14mu}{\sum\limits_{i}{\left\lbrack {\frac{\mathbb{d}n_{i}}{\mathbb{d}T} - {\left( {n_{i} - 1} \right) \cdot \left( {\alpha_{0} - \alpha_{i}} \right)}} \right\rbrack x_{i}}}} = 0}$

Finally, the cavity can comprise at least one material of which thecoefficient of variation of the optical index as a function oftemperature T is negative. Also, the cavity may comprise one or morethermal sensors and the laser gyro may comprise an electronic processingunit linked to said thermal sensors, which makes it possible tocalculate the residual variations as a function of temperature of thescale factor.

The invention also relates to an angular-velocity or angular-measurementsystem comprising at least one laser gyro as previously described. Thesystem can then comprise three laser gyros whose cavities are orientedso as to make measurements in three independent directions.

The invention will be better understood and other advantages will becomeapparent on reading the following description, given without limitation,and by virtue of the appended FIG. 1, which represents a basic diagramof a laser gyro according to the invention.

FIG. 1 represents the basic diagram of a laser gyro according to theinvention. It comprises:

-   -   a cavity 1 made of a first material and comprising several        reflecting mirrors 2, 3 and 4 and a partially reflecting mirror        5;    -   an amplifying medium 6;    -   at least one optical element 7 of length L₇;    -   the whole assembly being designed so that two optical waves can        propagate in two opposite directions inside the cavity. These        two waves are represented by a double line in FIG. 1. These        waves pass through the various optical elements arranged in the        cavity;    -   and an opto-electronic measurement device 8 represented by a        dashed line, which makes it possible to calculate the angular        parameter measured on the basis of the interference pattern of        the two counter-propagating waves coming from the partially        reflecting mirror 5.

As stated, the scale factor of the laser gyro S satisfies with the samenotation as previously: S=4A/λ·L

It is possible to replace the wavelength λ with the associated frequencyv. The new expression for the scale factor S is then obtained, which nowequals:

$S = \frac{4{A \cdot v}}{c \cdot L}$

The coefficient H, which equals:

$H = \frac{A}{L_{o}^{2}}$with L_(O) being the geometric perimeter of the cavity,

is called the aspect ratio.

H is a dimensionless parameter substantially independent of temperature.This is satisfied in particular if the external constraints experiencedby the cavity comply with the latter's symmetries. A is then replacedwith HL₀ ² in the expression for S to obtain:

$S = \frac{4{H \cdot L_{o}^{2} \cdot v}}{c \cdot L}$

The cavity comprises n optical elements Oi indexed, i being an integerindex varying between 1 and the total number n of optical elements, eachoptical element Oi being of length L_(i) and of optical index n_(i).Consequently the following relation holds:

$L = {L_{O} + {\sum\limits_{i}{\left( {n_{i} - 1} \right) \cdot L_{i}}}}$

The new expression for the scale factor S may then be written:

$S = \frac{4{H \cdot L_{O}^{2} \cdot v}}{c \cdot \left( {L_{O} + {\sum\limits_{i}{\left( {n_{i} - 1} \right)L_{i}}}} \right)}$

The stability condition for the scale factor as a function oftemperature may be written:

$\mspace{79mu}{{\frac{\mathbb{d}S}{\mathbb{d}T} = 0},{{{{{or}\left( {{2L_{0}\frac{\mathbb{d}L_{0}}{\mathbb{d}T}v} + {L_{0}^{2}\frac{\mathbb{d}v}{\mathbb{d}T}}} \right)}\left\lbrack {L_{0} + {\sum\limits_{i}{\left( {n_{i} - 1} \right)L_{i}}}} \right\rbrack} - {L_{0}^{2} \cdot v \cdot \left( {\frac{\mathbb{d}L_{0}}{\mathbb{d}T} + {\sum\limits_{i}{\left( {n_{i} - 1} \right)\frac{\mathbb{d}L_{i}}{\mathbb{d}T}}} + {\sum\limits_{i}{\frac{\mathbb{d}n_{i}}{\mathbb{d}T} \cdot L_{i}}}} \right)}} = 0}}$

which gives, after dividing by v·L₀ ³.

${{\left( {{\frac{2}{L_{0}}\frac{\mathbb{d}L_{0}}{\mathbb{d}T}} + {\frac{1}{v}\frac{\mathbb{d}v}{\mathbb{d}T}}} \right)\left\lbrack {1 + {\sum\limits_{i}{\left( {n_{i} - 1} \right)\frac{L_{i}}{L_{0}}}}} \right\rbrack} - {\frac{1}{L_{0}}\frac{\mathbb{d}L_{0}}{\mathbb{d}T}} - {\sum\limits_{i}{\frac{n_{i} - 1}{L_{0}}\frac{\mathbb{d}L_{i}}{\mathbb{d}T}}} - {\sum\limits_{i}{\frac{\mathbb{d}n_{i}}{\mathbb{d}T}\frac{L_{i}}{L_{0}}}}} = 0$

By putting

${x_{i} = \frac{L_{i}}{L_{0}}},$the following is obtained:

${{\left( {{\frac{2}{L_{0}}\frac{\mathbb{d}L_{0}}{\mathbb{d}T}} + {\frac{1}{v}\frac{\mathbb{d}v}{\mathbb{d}T}}} \right)\left\lbrack {1 + {\sum\limits_{i}{\left( {n_{i} - 1} \right)x_{i}}}} \right\rbrack} - {\frac{1}{L_{0}}\frac{\mathbb{d}L_{0}}{\mathbb{d}T}} - {\sum\limits_{i}{\frac{\left( {n_{i} - 1} \right)x_{i}}{L_{i}}\frac{\mathbb{d}L_{i}}{\mathbb{d}T}}} - {\sum\limits_{i}{\frac{\mathbb{d}n_{i}}{\mathbb{d}T}x_{i}}}} = 0$

The expansion coefficients of the various optical elements equal:

$\alpha_{i} = {\frac{1}{L_{i}}{\frac{\mathbb{d}L_{i}}{\mathbb{d}T}.}}$By introducing them into the above expression, the following isobtained:

${{\left( {{2\alpha_{0}} + {\frac{1}{v}\frac{\mathbb{d}v}{\mathbb{d}T}}} \right)\left\lbrack {1 + {\sum\limits_{i}{\left( {n_{i} - 1} \right)x_{i}}}} \right\rbrack} - \alpha_{0} - {\sum\limits_{i}{\left( {n_{i} - 1} \right)x_{i}\alpha_{i}}} - {\sum\limits_{i}{\frac{\mathbb{d}n_{i}}{\mathbb{d}T}x_{i}}}} = 0$

which is indeed the expression as claimed, called relation 1.

In the simple case where the cavity comprises only a single opticalelement, which serves as amplifying medium, said element having anoptical index n, a total length L and an expansion coefficient α, theabove relation may now be written:

${{\left( {{2\alpha_{0}} + {\frac{1}{v}\frac{\mathbb{d}v}{\mathbb{d}T}}} \right)\left\lbrack {1 + {\left( {n - 1} \right)x}} \right\rbrack} - \alpha_{0} - {\left( {n - 1} \right)\alpha\; x} - {\frac{\mathbb{d}n}{\mathbb{d}T}x}} = 0$

In this case, x therefore equals:

$x = \frac{\alpha_{0} + {\frac{1}{v}\frac{\mathbb{d}v}{\mathbb{d}T}}}{\frac{\mathbb{d}n}{\mathbb{d}T} + {\left( {n - 1} \right)\left( {\alpha - {2\alpha_{0}} - {\frac{1}{v}\frac{\mathbb{d}v}{\mathbb{d}T}}} \right)}}$

By way of example, if the amplifying medium is neodymium-YAG working atthe wavelength of 1.06 microns, then

-   -   the optical index n is equal to 1.82;    -   the linear expansion coefficient α is equal to 7.6 ppm per        degree, ppm signifying parts per million;    -   the coefficient of variation of the optical index as a function        of temperature is equal to 7.3 ppm per degree;    -   the coefficient of variation of the frequency

$\frac{1}{v}\frac{\mathbb{d}v}{\mathbb{d}T}$is equal to −4.6 ppm per degree,

-   -   and x represents the percentage of amplifying medium present in        the cavity.    -   In this case, x is equal to:

$x = \frac{\alpha_{0} - 4.6}{17.3 - {1.64\alpha_{0}}}$

x must lie between 0 and 1. Consequently, it suffices that thecoefficient of expansion of the material α₀ should satisfy:

4.6 ppm·K⁻¹<α₀.<8.3 ppm·K⁻¹ for it to be it possible to find a suitablelength of neodymium-YAG enabling the scale factor to be made almostindependent of temperature.

The above relation 1 makes it possible to preserve a constant scalefactor. For certain applications, it may be beneficial to avoiddisturbing the operation of the laser gyro by mode hops and to preservea temperature-independent emitted mode number, thus yielding thefollowing mathematical relation:

${{\frac{1}{v} \cdot \frac{\mathbb{d}v}{\mathbb{d}T}} + {\frac{1}{L} \cdot \frac{\mathbb{d}L}{\mathbb{d}T}}} = 0$

By using the same notation as previously, this relation may also bewritten:

${{\left\lbrack {L_{0} + {\sum\limits_{i}{\left( {n_{i} - 1} \right)L_{i}}}} \right\rbrack\frac{1}{v}\frac{\mathbb{d}v}{\mathbb{d}T}} + {\frac{\mathbb{d}L_{0}}{\mathbb{d}T}L_{i}} + {\sum\limits_{i}{\frac{\mathbb{d}n_{i}}{\mathbb{d}T}L_{i}}} + {\sum\limits_{i}{\left( {n_{i} - 1} \right)\frac{\mathbb{d}L_{i}}{\mathbb{d}T}}}} = 0$

which becomes, after dividing by L₀,

${{\left\lbrack {1 + {\sum\limits_{i}{\left( {n_{i} - 1} \right)x_{i}}}} \right\rbrack\frac{1}{v}\frac{\mathbb{d}v}{\mathbb{d}T}} + \alpha_{0} + {\sum\limits_{i}{\frac{\mathbb{d}n_{i}}{\mathbb{d}T}x_{i}}} + {\sum\limits_{i}{\left( {n_{i} - 1} \right)x_{i}\alpha_{i}}}} = 0$

The latter relation is called relation 2.

The conditions for relations 1 and 2 to be simultaneously satisfied are:

$\left\{ {\begin{matrix}{\alpha_{0} = {{- \frac{1}{v}} \cdot \frac{\mathbb{d}v}{\mathbb{d}T}}} \\{{\sum\limits_{i}{\left\lbrack {\frac{\mathbb{d}n_{i}}{\mathbb{d}T} - {\left( {n_{i} - 1} \right)\left( {\alpha_{0} - \alpha_{i}} \right)}} \right\rbrack x_{i}}} = 0}\end{matrix}\quad} \right.$

These conditions are called relations 3 and 4.

If the cavity comprises only a single optical element, relations 3 and 4above cannot necessarily be satisfied. If the cavity comprises at leasttwo optical elements, it becomes easier to satisfy these two relationssimultaneously. Indeed, with two optical elements, relations 3 and 4 maybe written:

$\left\{ {\begin{matrix}{\alpha_{0} = {{- \frac{1}{v}} \cdot \frac{\mathbb{d}v}{\mathbb{d}T}}} \\{{{\left\lbrack {\frac{\mathbb{d}n_{i}}{\mathbb{d}T} - {\left( {n_{1} - 1} \right)\left( {\alpha_{0} - \alpha_{1}} \right)}} \right\rbrack x_{1}} + {\left\lbrack {\frac{\mathbb{d}n_{2}}{\mathbb{d}T} - {\left( {n_{2} - 1} \right)\left( {\alpha_{0} - \alpha_{2}} \right)}} \right\rbrack x_{2}}} = 0}\end{matrix}\quad} \right.$

By way of example, if material 1 is Nd:YAG, serving as amplifyingmedium, conditions 3 and 4 become:

$\left\{ {\begin{matrix}{\alpha_{0} = {4.6\;{{ppm} \cdot K^{- 1}}}} \\{{{9.76 \cdot x_{1}} + {\left\lbrack {\frac{\mathbb{d}n_{2}}{\mathbb{d}T} - {\left( {n_{2} - 1} \right)\left( {\alpha_{0} - \alpha_{2}} \right)}} \right\rbrack x_{2}}} = 0}\end{matrix}\quad} \right.$

It is possible to find a material satisfying condition 3 since thisexpansion coefficient is typical of standard borosilicate-type glasses.

x₁ and x₂ necessarily being positive, relation 4 implies that:

${\frac{\mathbb{d}n_{2}}{\mathbb{d}T} - {\left( {n_{2} - 1} \right)\left( {4.6 - \alpha_{2}} \right)}} < 0$

This condition can be fulfilled for certain glasses. By way ofnon-limiting example, the characteristics of the reference glass PK51Afrom the Schott company are:α₂=12.7 ppm·K⁻¹dn ₂ /dT=−7 ppm·K⁻¹n₂=1.5

Also, consequently,

${{\frac{\mathbb{d}n_{2}}{\mathbb{d}T} - {\left( {n_{2} - 1} \right)\left( {4.6 - \alpha_{2}} \right)}} = {{- 3}\;{{ppm} \cdot K^{- 1}}}},$thus satisfying the above condition.

With this reference glass PK51A, relation 4 may then be written:9.76·x ₁−3·x ₂=0or x ₁ /x ₂=0.3

For example, if the YAG rod used for the amplification has a length of 5centimeters, then a 16-centimeter rod of PK51β permits thermalcompensation, the cavity being itself made of a material whose expansioncoefficient equals 4.6 ppm·K⁻¹, which is typically representative of aglass.

Thus, a judicious choice of materials makes it possible to compensate inlarge part for the thermal drift, acting on the scale factor and also onthe optical modes.

Of course, in the case where the expansion coefficients or thecoefficients of variation of the optical index as a function oftemperature are not perfectly linear, and in the case also where thetemperature is not uniformly distributed in the cavity enclosure, it ispossible to refine the measurement of the laser gyro scale factor bymeans of a mathematical model that determines the small variations inthe scale factor as a function of thermal variations. In such a case,temperature sensors are placed in the cavity enclosure.

The laser gyro according to the invention can apply to anyangular-velocity or angular-measurement system. The system can comprise,in particular, three laser gyros whose cavities are oriented so as tomake measurements in three different directions, thus making it possibleto ascertain the three angular components of a position or of avelocity.

1. A laser gyro, comprising: a ring cavity of optical length L and ofarea A, and a solid-state amplifying medium wherein two optical waves ofaverage wavelength λ propagate in opposite directions inside the cavity,the scale factor S of the laser gyro being equal to$\frac{4 \cdot A}{\lambda \cdot L},$ wherein the cavity has a geometricperimeter L_(o), said cavity comprising at least one optical element Oitraversed by the optical waves, i being an integer index varying between1 and the total number of optical elements, the optical element Oihaving length L_(i), optical index n_(i), x_(i) being equal to the ratio$\frac{L_{i}}{L_{O}},\frac{\mathbb{d}n_{i}}{\mathbb{d}T}$ being thecoefficient of variation as a function of temperature T of the opticalindex n_(i) of the optical element Oi, α_(i) being the linear expansioncoefficient of the optical element Oi, α₀ being the linear expansioncoefficient of the constituent material of the cavity, v being thecentral emission frequency of the amplifying medium and$\frac{\mathbb{d}v}{\mathbb{d}T}$ being the coefficient of variation asa function of temperature T of said frequency, said optical elements andthe amplifying medium are designed so that the following relation issubstantially satisfied:${{\left( {{2 \cdot \alpha_{0}} + {\frac{1}{v} \cdot \frac{\mathbb{d}v}{\mathbb{d}T}}} \right)\left\lbrack {1 + {\sum\limits_{i}{\left( {n_{i} - 1} \right)x_{i}}}} \right\rbrack} - \alpha_{0} - {\sum\limits_{i}{\left( {n_{i} - 1} \right){x_{i} \cdot \alpha_{i}}}} - {\sum\limits_{i}{\frac{\mathbb{d}n_{i}}{\mathbb{d}T}x_{i}}}} = 0.$2. The laser gyro as claimed in claim 1, wherein the optical elementsand the amplifying medium are designed so that the following relationsare substantially and simultaneously satisfied:$\alpha_{0} = {{- \frac{1}{v}} \cdot \frac{\mathbb{d}v}{\mathbb{d}T}}$and${\sum\limits_{i}\left\lbrack {\frac{\mathbb{d}n_{i}}{\mathbb{d}T} - {\left( {n_{i} - 1} \right) \cdot \left( {\alpha_{0} - \alpha_{i}} \right)}} \right\rbrack} = {x_{i} = 0.}$3. The laser gyro as claimed in claim 1, wherein the cavity comprises atleast one material of which the coefficient of variation of the opticalindex as a function of temperature T is negative.
 4. The laser gyro asclaimed in claim 1, wherein the cavity comprises at least one thermalsensor and the laser gyro comprises an associated electronic processingunit linked to said thermal sensor, which makes it possible to calculatethe residual variations as a function of temperature of the scalefactor.
 5. An angular-measurement system, comprising one ring cavity ofoptical length L and of area A, and a solid-state amplifying medium thatare designed in such a way that two optical waves of average wavelengthλ propagate in opposite directions inside the cavity, the scale factor Sof the laser gyro being equal to $\frac{4 \cdot A}{\lambda \cdot L},$wherein as the cavity has a geometric perimeter L_(o), said cavitycomprising at least one optical element traversed by the optical waves,i being an integer index varying between 1 and the total number ofoptical elements, the optical element Oi being of length L_(i), ofoptical index n_(i), x_(i) being equal to the ratio$\frac{L_{i}}{L_{O}},\frac{\mathbb{d}n_{i}}{\mathbb{d}T}$ being thecoefficient of variation as a function of temperature T of the opticalindex n_(i) of the optical element Oi, α_(i) being the linear expansioncoefficient of the optical element Oi, α₀ being the linear expansioncoefficient of the constituent material of the cavity, v being thecentral emission frequency of the amplifying medium and$\frac{\mathbb{d}v}{\mathbb{d}T}$ being the coefficient of variation asa function of temperature T of said frequency, said optical elements andthe amplifying medium are designed so that the following relation issubstantially satisfied:${{\left( {{2 \cdot \alpha_{0}} + {\frac{1}{v} \cdot \frac{\mathbb{d}v}{\mathbb{d}T}}} \right)\left\lbrack {1 + {\sum\limits_{i}{\left( {n_{i} - 1} \right)x_{i}}}} \right\rbrack} - \alpha_{0} - {\sum\limits_{i}{\left( {n_{i} - 1} \right){x_{i} \cdot \alpha_{i}}}} - {\sum\limits_{i}{\frac{\mathbb{d}n_{i}}{\mathbb{d}T}x_{i}}}} = 0.$6. The measurement system as claimed in claim 5, comprising three lasergyros whose cavities are oriented so as to make measurements in threedifferent directions.
 7. The laser gyro as claimed in claim 1, whereinthe cavity comprises at least one material of which the coefficient ofvariation of the optical index as a function of temperature T isnegative.
 8. The laser gyro as claimed in claim 1, wherein the cavitycomprises at least one thermal sensor and the laser gyro comprises anassociated electronic processing unit linked to said thermal sensor,which makes it possible to calculate the residual variations as afunction of temperature of the scale factor.
 9. The laser gyro asclaimed in claim 2, wherein the cavity comprises at least one thermalsensor and the laser gyro comprises an associated electronic processingunit linked to said thermal sensor, which makes it possible to calculatethe residual variations as a function of temperature of the scalefactor.